What pattern on my film would I see if I expose it to the Sun at noon everyday for a year?

This answer is courtesy of Louis Strous of the
National Solar Observatory, Sacramento Peak, NM.

If you expose film to the Sun at noon every day for a year, then what
path the Sun follows in the picture depends on which noon you mean.

Before there were accurate clocks, noon meant "high noon": the time
when the Sun was due south or highest in the sky (which is practically
the same thing). If you take your pictures at that noon every day,
then the path that the Sun follows in the picture is a straight
vertical line.

The Sun moves up and down (north and south) along that line with the
seasons: at midsummer it is highest, and at midwinter it is lowest.
This happens because the rotation axis of the Earth (the line
connecting the north and south poles) is not perpendicular to the
direction to the Sun, so that during one half of the year (from March
to September) the north pole is closer to the Sun than the south pole,
and during the other half of the year (from September to March) the
south pole is closer than the north pole. When the closest pole to
you moves closer to the Sun (relative to the center of the Earth),
then the Sun moves higher in your sky. When the closest pole moves
further away from the Sun then the Sun moves lower in your sky.
Between December and June the Sun moves north in the sky, and between
June and December it moves south. The tilt angle over which the poles
can get closer to or further away from the Sun is called the *obliquity
of the ecliptic* and is about 23 degrees.

If you mean noon on the clock (excluding daylight savings time), then
the Sun will follow a path in your picture with a shape like the
number 8. This path is called the *analemma*. It consists of the
same up-and-down motion as in the previous case, but now also has
motion to the left and right (east and west). This left-right motion
has two main causes.

If the Sun traveled in the east-west direction through our sky at a
constant speed, then it would keep perfect pace with the clock, and
then there would not be any left-right component to the analemma.
Let's call such a Sun the *mean Sun*.

Because of the obliquity of the ecliptic, the Sun moves not just in
the east-west direction in the sky, but also in the north-south
direction. This means that even if the Sun moved along the sky at a
constant rate, its speed in the east-west direction would vary because
its speed in the north-south direction varies, so that the real Sun
would go faster in the east-west direction than the mean Sun near
midsummer and midwinter, and slower near the beginning of spring and
autumn. That is why the Sun moves along the analemma from west to
east near the top and bottom of the analemma (around midwinter and
midsummer, December and June), and from east to west near the middle
of the analemma (around the beginning of spring and autumn, March and
September).

The Earth's orbit is an ellipse, which is a circle that is slightly
squashed in one direction. Because the Earth's orbit around the Sun
is not a circle, the apparent speed of the Sun in the sky varies
slightly with time. Around 3 January it is greatest (about 3 percent
faster than average), and around 7 July it is smallest (about 3
percent slower than average). This adds to the east-west motion in
the analemma. Because the apparent speed of the Sun in the sky is
faster in December than in June, the southern (September to March)
part of the analemma is wider than the northern (March to September)
part.

The analemma is 47 degrees long (two times the obliquity of the
ecliptic) and, at its widest part, about 8 degrees wide.